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In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Other generating functions of random variables include the moment-generating function, the characteristic function and the cumulant generating function. The probability generating function is also equivalent to the factorial moment generating function , which as E [ z X ] {\displaystyle \operatorname {E} \left[z^{X}\right]} can also be ...
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and ...
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable X is defined as = [] for all complex numbers t for which this expected value exists.
For a degenerate point mass at c, the cumulant generating function is the straight line () =, and more generally, + = + if and only if X and Y are independent and their cumulant generating functions exist; (subindependence and the existence of second moments sufficing to imply independence.
The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above, =, and substitute the expression for P from equation , expressed in terms of p and Q: